A Derivation of Titius-Bode Type Relations for the Planets of the Solar System
and Satellite Systems of the Planets

This is a derivation of the relationships of orbit radii and periods of revolution to the order number
of the orbits for the
planets and their satellite systems. Ever since 1768 when Johann Bode published the remarkable empirical relationship for distances
of the planets from the Sun astronomers have puzzled over whether there is any physical justification
for it. Bode did not originate the relationship; he only publicized it. Johann Titius in 1766 had
earlier stated it in a footnote of his translation of a 1764 book by Charles Bonnet. Apparently the
earliest statement of the relationship was by David Gregory in 1715. The relationship of the
orbit radii of the planets given as ratios to that of Earth's orbit radius is:

R1 = 0.4
Rn = 0.4 + 0.3*2n-2

where n is the order number of the orbit starting with n=1 for Mercury. The relationship posits a
planetary orbit number 5 where the asteroid belt is.

The empirical fit is quite striking.

Mercury

Venus

Earth

Mars

AsteroidBelt

Jupiter

Saturn

Uranus

Neptune/Pluto

OrderNumber

1

2

3

4

5

6

7

8

9

Titius-Bode Law

0.4

0.7

1.0

1.6

2.8

5.2

10.0

19.6

39

ActualValue

0.387

0.723

1.0

1.524

2.7*

5.203

9.539

19.18

30.06/39.52

*Ceres

It is often said that there is no theoretical basis for this relationship. While the form
Rn=c+a*bn does not
have a theoretical justification there is justification of a relationship of the form Rn=a*bn. This will be given below.
For the solar system and some of the satellites of Jupiter, Saturn and Uranus each planet or satellite
is roughly twice as far from the system center as the preceding satellite. (For more on the
relationships which exists for the satellite systems of Jupiter, Saturn and
Uranus see Bode2.)
There has to be a physical explanation for this pattern. And there is. The explanation is in terms of
resonance.

The explanation is not in terms of radii per se; it is in terms of orbit periods.
There is of course a relationship between orbit period and orbit radius, called Kepler's
Law, which says the cube of orbit radius is proportional to the square of orbit period. The physical
process accounting for the radii at which planets formed involves their orbit periods.

The original system was a planetary ring rotating about the Sun not as a unit but instead with Kepler's
Law satisfied. Once one planet was formed any planetary material having an orbit period equal to one half
the orbit period of the planet would be nudged out of its orbit. This is the phenomenon of
resonance.
If any two bits of matter are in orbits such that one makes two revolution about the Sun for every one
the other makes then their gravitational fields will ultimately nudge each other out of those resonant
orbits. They do not have to move very far to break the resonance.

It is a remarkable property of resonance that the ultimate effect does not depend upon the magnitude of the
perturbance but instead only on matching of frequencies.

The phenomenon of resonance occurs both ways; toward an inner band and toward an outer band. For purposes
of explanation it is convenient to focus on the inner resonance band created by a planet.

Once the material's orbit period was significantly different from one half of the planet's orbit period
the resonance would be broken. Resonance would also occur if the planetary material orbited the Sun three
times for every two times the planet orbited it. Likewise for the material having an orbit period 2/5 of
the planet's orbit period.

The significant feature of resonance is that the disturbing influence does not have to be strong to
eventually have a great effect; it only has to be of the right frequency.

It has long be recognized that resonance has had an important role in the formation of the structure of the solar
system. For example, among the asteroids there are none which have periods which are 1/3, 1/2 and 2/5 of
that of Jupiter. The absence of asteroids at or near the resonance points are what are known as the
Kirkwood Gaps, named after the American astronomer Daniel Kirkwood who discovered the phenomenon in 1886.
Here is the plot of the orbit size of about 157 thousand asteroids.

Note that the frequency rises to peaks near the resonance points.

In the rings of Saturn there is a gap of 1700 miles that corresponds to 1/2 the
period of the moon Mimas, 1/3 the period of Enceladus and 1/4 the period of Tethys.

However it seemed paradoxical that planets are found close to the forbidden resonance bands.
This is no paradox. The planetary material did not have to move very far from the forbiddedn resonance
zone to break the resonance. However when the planetary material moved away from its original orbit
it would be moving at a velocity different from that of the surrounding material. This led to collisions
and agglomeration of material. As illustrated in the diagram
the planetary material would be concentrated in the space near the resonance band. Thus planets would form near
the resonance bands.

Planetary material close to the forming planet would be swept up by collision and by the gravitational field of the planet. The
planets acquired not only mass but angular momentum in this process. See
Planetary Sweep.

Here are the ratios of the orbit period of the next closer to the Sun to each planet's orbit period.
The ratio of the period for the planet next farther away from the Sun to the period for the planet is
called the Reciprocal Ratio in the table.

Planet orPlanetoid

Ratio

ReciprocalRatio

Mercury

---

2.557

Venus

0.391

1.623

Earth

0.616

1.900

Mars

0.526

2.421

Ceres

0.413

2.609

Jupiter

0.383

2.458

Saturn

0.407

2.847

Uranus

0.351

1.964

Neptune

0.509

1.503

Pluto

0.665

---

The average of the nine ratios given is 0.4736.
If Pluto is left out the average is 0.4496, notably close the average of 0.4 and 0.5.

Here is the tabulation
of the frequencies of the ratios for 0.02 intervals from 0.3 to 0.62. Pluto is left out of the tabulation.

The ratios are generally in the vicinity of 0.4 and 0.5 and the reciprocal ratios in the vicinity of
2.0 or 2.5. The cases
where the ratio is not near 0.4 or 0.5 are situations where something besides the ratio is unusual. Pluto is most likely a moon-like object
that escaped from its primary. Uranus is the odd case in which its angle of inclination is about 90°. This probably indicates that something
catastrophic happened to Uranus. The case of Venus and Earth being
so close together is a puzzle, but Venus has retrograde rotation, indicating also that something
catastrophic occurred. However even for the odd cases the ratios are close to a resonance.
Venus may be a case in
which the planet formed near the 3/5 resonance band. Likewise Uranus could have formed near the 1/3 resonance
band. And, of course, Pluto could have formed near the 3/2 resonance band for Neptune but there is always
the possibility that
Pluto simply should be left out of the analysis.

Planet orAsteroid

Ratio

ResonanceRatio

Deviation

ProportionalDeviation

Mercury

0.391

0.4

−0.009

−0.0225

Venus

0.616

0.6

+0.016

+0.027

Earth

0.526

0.5

+0.026

+0.052

Mars

0.413

0.4

+0.013

+0.0325

Ceres

0.383

0.4

−0.017

−0.0425

Jupiter

0.407

0.4

+0.007

+0.0175

Saturn

0.351

0.333

+0.018

+0.054

Uranus

0.509

0.5

+0.009

+0.018

The asteroid Ceres is used to represent the asteroid belt in the table.
However the ratio of the period for Mars to that of Jupiter is 0.158 and the square root of this
ratio is 0.398.

Here is the distribution of the proportional deviations.

The average of the proportional deviations from the resonance ratios is 0.017 and the average
of the absolute values of the proportional deviations is 0.03325.

Thus planets formed from material having an orbit period of approximately either 0.5 or 0.4
of that of the next outer planet. Whether it was near 0.5, 0.4 or another resonance ratio was a matter
of chance. In any case the end result for a sequence of planets is the
same as if the planets formed near the midpoint of the 0.4 and 0.5 resonance bands. This means that, on
average, the
orbit period of a planet would be roughly 0.45
of that of the next outer planet and that the outer planet would have an orbit period of
approximately 1/0.45=2.22 times that of the inner planet; i.e.,
the next closer planet to the Sun. Proto-planets might possibly
have formed near both the 0.5 and 0.4 resonance bands and later coalesced into a a single planet or
a planet and its satellite.

The prior observations are in terms of orbit periods. Since by Kepler's Law the corresponding orbit
radii are (0.5)2/3=0.63 and
(0.4)2/3=0.54.
The planetary material is
spread out over space. Thus, rather than averaging 0.4 and
0.5 for the period times it is more appropriate to average the 0.54 and 0.63 of distances. This gives 0.585, which
corresponds to an average orbit period of (0.585)3/2=0.44744. The reciprocal of
this value is 2.23494.

Thus according to the above analysis the ratios of the orbit periods of planets should average 2.23494.
In other words the relationship of orbit period to order number should be

Tn = a*bnwith b=2.23494

In logarithmic form the above relationship is

log10(Tn) = log10(a) + n*log10(b)

In the above graph it is seen that Pluto represents a deviation from the pattern for the planets but it
is not an extreme deviation.

The regression of the logarithms of orbit periods relative to that of Earth on order number for the
eight planets and Pluto gives

Tn = 0.10186*(2.235116)n

The empirical estimate of b=2.235116 is stunningly close to the value of b=2.23494 based upon planetary
material being excluded from the 0.4 and 0.5 resonance bands.

The above results concerning resonance band ratios suggests that it may be of interest to look at the
ratio of the period of the second next planet to that of the planet, as in the
case of Mars and Jupiter.

Planet orPlanetoid

Ratio

ReciprocalRatio

Mercury

---

4.1511

Venus

---

3.0844

Earth

0.2409

4.6

Mars

0.3242

6.3158

Ceres

0.2174

6.4130

Jupiter

0.1583

7.0000

Saturn

0.1559

5.5932

Uranus

0.1429

---

Neptune

0.1788

---

Pluto

0.3387

---

In the table Ceres was used as a representative of the asteroid belt. Four of the eight ratios are
in the vicinity of 0.16=(0.4)². One is close to 0.25=(0.5)² and one close of 0.2=(0.4)(0.5).
Only the ratios for Mars and Pluto are not a combination of 0.4 and
0.5. Their ratios of course corresponds to a 1/3 resonance.

The Exclusion of Pluto from the Analysis

The attempt to find a relationship that includes Pluto distorts the results unnecessarily. Making the
relationship fit Pluto
results in Uranus and Neptune not fitting very well. Leaving Pluto out of the regression results in
the equation

Tn = 0.094854*(2.280933)n

This is the relationship which will be used for further analysis.
Here is the comparison of the actual orbit periods and the values computed from the equation.

Planet

OrbitPeriod(years)

ComputedPeriod(years)

Mercury

0.2409

0.2164

Venus

0.616

0.4935

Earth

1.0000

1.1256

Mars

1.9

2.5675

Jupiter

12.0

13.3576

Saturn

29.5

30.4689

Uranus

84

69.4952

Neptune

165

158.514

For order number 5 the equation gives a period of 5.856 years, whereas the period for Ceres is 4.6 years.
For order number 10 the equation gives a period of 361.56 years, whereas the period for Pluto is 248 years.

The Relationship of the Planets' Orbit Radii to Their Order Number:
The Titius-Bode Type Relationship

When orbit radii and orbit periods are expressed relative the values for Earth Kepler's Law takes the
form

R = T2/3

Thus the corresponding relationship of planet orbit radius and order number is

Rn = 0.207987*(1.732773)n

Using the value of b=2.23494 would give a value of 1.7094 instead of 1.732773. This is
notably close.
Here is the comparison of the actual orbit radii and the values computed from the equation, both
expressed in Astronomical Units (A.U.), distance relative to Earth's orbit radius.

Planet

OrbitDistance(A.U.)

ComputedDistance(A.U.)

Mercury

0.387

0.3604

Venus

0.723

0.6245

Earth

1.0000

1.0821

Mars

1.524

1.8750

Jupiter

5.203

5.6297

Saturn

9.539

9.7550

Uranus

19.18

16.9032

Neptune

30.06

29.2895

For order number 10 the equation gives a distance of 50.75 whereas Pluto's distance is 39.5.
For order number 5 the equation gives a distance of 3.249, whereas the distance of Ceres is 2.7.

Other Satellite Systems

Because of the nature of the derivation this type of relation should apply for any
satellite system such as the moons of Jupiter, Saturn and Uranus. The corresponding
regression equations for these systems are

Although the regression coefficients for order number are significantly different from
the value of 0.35 found for the planets the values of the same order of magnitude and
reasonably close.

The Planetary Systems of Other Stars

As asserted before, the derivation would apply to any satellite system and that would
include the planetary systems of other stars. In the December 6, 2008 issue of Science News
it was reported that images of three planets were found for the star HR8799 which lies about
130 light-years from our solar system. The planets are massive, one having 10 times the mass of Jupiter.

These planets lie 25, 40 and 70 astronomical units (A.U.) from their star.
This means that, by Kepler's Law, their orbit periods are proportional to 253/2, 403/2 and 703/2; i.e.,
125, 253 and 585.7. The constant of proportionality
depends on the mass of the star relative to that of the Sun. This constant however does not affect their ratios.
The ratio of the period of the middle planet to the outer planet is 0.432. The ratio of the
period of the inner planet to that of the middle planet is 0.494. Thus the planets are located
near the 0.4 and 0.5 resonance bands. This is essentially the same pattern as prevails for the
planets of our solar system.

Conclusion

There is a physical justification for a Titius-Bode relationship between the order number
of a planet and its orbit radius, but that relationship is derived from the relationship of
order number and orbit period. The planets formed near the resonance bands as a result of
planetary material being nudged away from the resonance band.
When planetary material
was nudged away from a resonance band its velocity differed from that of the material in its new location.
The subsequent collisions aggregated the material into a proto-planet that continued to sweep up material
in its band and nearby bands due to the burgeoning gravitational field of the proto-planet.
Thus the planets
formed near the resonance bands, either above or below just due to chance. Possibly two or more proto-planets may have
formed both above and below the resonance band and then later coalesced into one planet at their center
of gravity which would have been close to the resonance band.

The resonance bands near which
planets were formed were usually the 0.4 and 0.5 resonance bands and the particular one was
a matter of chance. On average the planets were formed such that the ratio of the orbit period of
the next innermost planet to the orbit period of the planet is about 0.45.

When the orbit periods and orbit radii are expressed relative to those of Earth the relationships
are:

Tn = 0.094854*(2.280933)nand
Rn = 0.207987*(1.732773)n

The evidence from the satellite systems of Jupiter, Saturn and Uranus and also the star HR 8799 is
that not only do the different systems have Titius-Bode type relationships between the orbit
periods and orbit radii and the order numbers of the satellites but that it is approximately
the same relationship.